A306484 Expansion of Product_{k>=1} 1/(1 - Lucas(k)*x^k), where Lucas = A000204.

1, 1, 4, 8, 24, 47, 129, 255, 641, 1308, 3064, 6225, 14286, 28792, 63571, 129240, 278329, 561044, 1190501, 2387695, 4987250, 9976529, 20536591, 40879937, 83416195, 165182927, 333581057, 658385847, 1318764282, 2590568669, 5154370637, 10082762399, 19929958391, 38848175389, 76331335061, 148233818041

Offset: 0

Author

Ilya Gutkovskiy, Feb 18 2019

Keywords

• nonn

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..4000

Crossrefs

Cf. A000204, A261031, A300520.

Programs

Mathematica

nmax = 35; CoefficientList[Series[Product[1/(1 - LucasL[k] x^k), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 35; CoefficientList[Series[Exp[Sum[Sum[LucasL[j]^k x^(j k)/k, {j, 1, nmax}], {k, 1, nmax}]], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d LucasL[d]^(k/d), {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 35}]

Formula

G.f.: exp(Sum_{k>=1} Sum_{j>=1} Lucas(j)^k*x^(j*k)/k).

From Vaclav Kotesovec, Feb 23 2019: (Start)

a(n) ~ c * 3^(n/2), where

c = 27050904.849254721356174679220734831574107371522481898944915... if n is even,

c = 27050894.152054775323471273913497954429537332266942696921416... if n is odd.

In closed form, c = ((3 + sqrt(3)) * Product_{k>=3}(1/(1 - Lucas(k)/3^(k/2))) + (-1)^n * (3 - sqrt(3)) * Product_{k>=3}(1/(1 - (-1)^k*Lucas(k)/3^(k/2))))/4.

(End)